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Does the polynomial hierarchy collapse if onto functions are invertible?

Journal

Theory of Computing Systems

Volume | Issue number

46 | 1

Pages (from-to)

143-156

Document type

Article

Faculty

Interfacultary Research Institutes

Institute

Institute for Logic, Language and Computation (ILLC)

Abstract

The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially
verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in
polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued function in deterministic
polynomial-time we mean on every input producing one of the possible values of the function on that input.We give a relativized
negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomial-time
hierarchy is infinite. We also show that relative to this same oracle, P not equal UP and TFNP(NP) functions are not computable
in polynomial-time with an NP oracle.

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